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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.relative_error"></a><a class="link" href="relative_error.html" title="Relative Error">Relative Error</a>
</h2></div></div></div>
<p>
      Given an actual value <span class="emphasis"><em>a</em></span> and a found value <span class="emphasis"><em>v</em></span>
      the relative error can be calculated from:
    </p>
<div class="blockquote"><blockquote class="blockquote"><p>
        <span class="inlinemediaobject"><img src="../../equations/error2.svg"></span>

      </p></blockquote></div>
<p>
      However the test programs in the library use the symmetrical form:
    </p>
<div class="blockquote"><blockquote class="blockquote"><p>
        <span class="inlinemediaobject"><img src="../../equations/error1.svg"></span>

      </p></blockquote></div>
<p>
      which measures <span class="emphasis"><em>relative difference</em></span> and happens to be less
      error prone in use since we don't have to worry which value is the "true"
      result, and which is the experimental one. It guarantees to return a value
      at least as large as the relative error.
    </p>
<p>
      Special care needs to be taken when one value is zero: we could either take
      the absolute error in this case (but that's cheating as the absolute error
      is likely to be very small), or we could assign a value of either 1 or infinity
      to the relative error in this special case. In the test cases for the special
      functions in this library, everything below a threshold is regarded as "effectively
      zero", otherwise the relative error is assigned the value of 1 if only
      one of the terms is zero. The threshold is currently set at <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;&gt;::</span><span class="identifier">min</span><span class="special">()</span></code>: in other words all denormalised numbers
      are regarded as a zero.
    </p>
<p>
      All the test programs calculate <span class="emphasis"><em>quantized relative error</em></span>,
      whereas the graphs in this manual are produced with the <span class="emphasis"><em>actual error</em></span>.
      The difference is as follows: in the test programs, the test data is rounded
      to the target real type under test when the program is compiled, so the error
      observed will then be a whole number of <span class="emphasis"><em>units in the last place</em></span>
      either rounded up from the actual error, or rounded down (possibly to zero).
      In contrast the <span class="emphasis"><em>true error</em></span> is obtained by extending the
      precision of the calculated value, and then comparing to the actual value:
      in this case the calculated error may be some fraction of <span class="emphasis"><em>units in
      the last place</em></span>.
    </p>
<p>
      Note that throughout this manual and the test programs the relative error is
      usually quoted in units of epsilon. However, remember that <span class="emphasis"><em>units
      in the last place</em></span> more accurately reflect the number of contaminated
      digits, and that relative error can <span class="emphasis"><em>"wobble"</em></span>
      by a factor of 2 compared to <span class="emphasis"><em>units in the last place</em></span>.
      In other words: two implementations of the same function, whose maximum relative
      errors differ by a factor of 2, can actually be accurate to the same number
      of binary digits. You have been warned!
    </p>
<h5>
<a name="math_toolkit.relative_error.h0"></a>
      <span class="phrase"><a name="math_toolkit.relative_error.zero_error"></a></span><a class="link" href="relative_error.html#math_toolkit.relative_error.zero_error">The
      Impossibility of Zero Error</a>
    </h5>
<p>
      For many of the functions in this library, it is assumed that the error is
      "effectively zero" if the computation can be done with a number of
      guard digits. However it should be remembered that if the result is a <span class="emphasis"><em>transcendental
      number</em></span> then as a point of principle we can never be sure that the
      result is accurate to more than 1 ulp. This is an example of what <a href="http://en.wikipedia.org/wiki/William_Kahan" target="_top">http://en.wikipedia.org/wiki/William_Kahan</a>
      called <a href="http://en.wikipedia.org/wiki/Rounding#The_table-maker.27s_dilemma" target="_top">http://en.wikipedia.org/wiki/Rounding#The_table-maker.27s_dilemma</a>:
      consider what happens if the first guard digit is a one, and the remaining
      guard digits are all zero. Do we have a tie or not? Since the only thing we
      can tell about a transcendental number is that its digits have no particular
      pattern, we can never tell if we have a tie, no matter how many guard digits
      we have. Therefore, we can never be completely sure that the result has been
      rounded in the right direction. Of course, transcendental numbers that just
      happen to be a tie - for however many guard digits we have - are extremely
      rare, and get rarer the more guard digits we have, but even so....
    </p>
<p>
      Refer to the classic text <a href="http://docs.sun.com/source/806-3568/ncg_goldberg.html" target="_top">What
      Every Computer Scientist Should Know About Floating-Point Arithmetic</a>
      for more information.
    </p>
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